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A028895 5 times triangular numbers: a(n) = 5*n*(n+1)/2.

%I A028895 #105 Aug 07 2025 14:46:54
%S A028895 0,5,15,30,50,75,105,140,180,225,275,330,390,455,525,600,680,765,855,
%T A028895 950,1050,1155,1265,1380,1500,1625,1755,1890,2030,2175,2325,2480,2640,
%U A028895 2805,2975,3150,3330,3515,3705,3900,4100,4305,4515,4730,4950,5175,5405,5640
%N A028895 5 times triangular numbers: a(n) = 5*n*(n+1)/2.
%C A028895 Sequence found by reading the line from 0, in the direction 0, 5, ... and the same line from 0, in the direction 0, 15, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Axis perpendicular to A195142 in the same spiral. - _Omar E. Pol_, Sep 18 2011
%C A028895 Bisection of A195014. Sequence found by reading the line from 0, in the direction 0, 5, ..., and the same line from 0, in the direction 0, 15, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. This is the main diagonal of the spiral. - _Omar E. Pol_, Sep 25 2011
%C A028895 a(n) = the Wiener index of the graph obtained by applying Mycielski's construction to the complete graph K(n) (n>=2). - _Emeric Deutsch_, Aug 29 2013
%C A028895 Sum of the numbers from 2*n to 3*n for n=0,1,2,... - _Wesley Ivan Hurt_, Nov 27 2015
%C A028895 Numbers k such that the concatenation k625 is a square, where also 625 is a square. - _Bruno Berselli_, Nov 07 2018
%C A028895 From _Paul Curtz_, Nov 29 2019: (Start)
%C A028895 Main column of the pentagonal spiral for n (A001477):
%C A028895                          50
%C A028895                       49 30 31
%C A028895                    48 29 15 16 32
%C A028895                 47 28 14  5  6 17 33
%C A028895              46 27 13  4  0  1  7 18 34
%C A028895                45 26 12  3  2  8 19 35
%C A028895                  44 25 11 10  9 20 36
%C A028895                    43 24 23 22 21 37
%C A028895                     42 41 40 39 38
%C A028895 (End)
%D A028895 D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 205.
%H A028895 Ivan Panchenko, <a href="/A028895/b028895.txt">Table of n, a(n) for n = 0..1000</a>
%H A028895 Rangaswami Balakrishnan and S. Francis Raj, <a href="http://dx.doi.org/10.7151/dmgt.1509">The Wiener number of powers of the Mycielskian</a>, Discussiones Math. Graph Theory, Vol. 30, No. 3 (2010), pp. 489-498 (see Theorem 2.1).
%H A028895 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A028895 G.f.: 5*x/(1-x)^3.
%F A028895 a(n) = 5*n*(n+1)/2 = 5*A000217(n).
%F A028895 a(n+1) = 5*n+a(n). - _Vincenzo Librandi_, Aug 05 2010
%F A028895 a(n) = A005891(n) - 1. - _Omar E. Pol_, Oct 03 2011
%F A028895 a(n) = A130520(5n+4). - _Philippe Deléham_, Mar 26 2013
%F A028895 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. - _Wesley Ivan Hurt_, Nov 27 2015
%F A028895 a(n) = Sum_{i=0..n} A001068(4i). - _Wesley Ivan Hurt_, May 06 2016
%F A028895 E.g.f.: 5*x*(2 + x)*exp(x)/2. - _Ilya Gutkovskiy_, May 06 2016
%F A028895 a(n) = A055998(3*n) - A055998(2*n). - _Bruno Berselli_, Sep 23 2016
%F A028895 From _Amiram Eldar_, Feb 26 2022: (Start)
%F A028895 Sum_{n>=1} 1/a(n) = 2/5.
%F A028895 Sum_{n>=1} (-1)^(n+1)/a(n) = (2/5)*(2*log(2) - 1). (End)
%F A028895 Product_{n>=1} (1 - 1/a(n)) = -(5/(2*Pi))*cos(sqrt(13/5)*Pi/2). - _Amiram Eldar_, Feb 21 2023
%p A028895 [seq(5*binomial(n,2), n=1..45)]; # _Zerinvary Lajos_, Nov 24 2006
%t A028895 Table[Sum[i + 2*n - 1, {i, 2, n}], {n, 45}] (* _Zerinvary Lajos_, Jul 11 2009 *)
%t A028895 Table[5 n (n + 1)/2, {n, 0, 50}] (* _Bruno Berselli_, Sep 23 2016 *)
%t A028895 5 Accumulate[Range[0,60]] (* _Harvey P. Dale_, Jun 17 2025 *)
%o A028895 (Magma) [5*n*(n+1)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Jun 09 2014
%o A028895 (PARI) a(n)=5*n*(n+1)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%o A028895 (Python)
%o A028895 def A028895(n): return 5*n*(n+1)>>1 # _Chai Wah Wu_, Aug 07 2025
%Y A028895 Cf. A000217, A001068, A005891, A028896, A046092, A055998, A085787, A130520, A195013, A195014, A195142.
%Y A028895 Cf. index to numbers of the form n*(d*n+10-d)/2 in A140090.
%Y A028895 Cf. A000566, A005475, A005476, A033583, A085787, A147875, A192136, A326725 (all in the spiral).
%K A028895 nonn,easy
%O A028895 0,2
%A A028895 Joe Keane (jgk(AT)jgk.org), Dec 11 1999